The following built-in MATLAB functions and commands are permitted for this assignment.
Vector/Matrix
Flow Control
Strings and Character Arrays
Other
Points will be deducted from any programs using functions outside this list.
Section Homework 5
Before you begin ...
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DO NOT use
AIfor any part of this problem. -
Review the Homework Policies before you begin.
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Follow the Submitting Work before you begin.
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Do not use any built-in MATLAB commands unless specified below.
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Exercises labeled as a βFunctionβ, should use the Function Template.
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Exercises labeled as a βScriptβ, should use the Script Template.
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Name the file
function_name.m, where eachfunction_nameis listed below. -
Upload each of these files to the homework page on Canvas. No zip files.
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The due date is specified on Canvas
Permitted MATLAB Functions & Commands for Homework 5.
Summary.
The focus of this assignment is on approximating different quantities using the concept of Monte Carlo Simulation.
Subsection pHat_marginErr_w_CL
This helper function computes the margin of error for a sample proportion estimate \(\hat{p}\) given the number of trials \(N\) and a desired confidence level.
Inputs:
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N (1x1) integer β number of samples or trials.
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Outputs:
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Details:
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βΈ The margin of error formula is \(\text{ME} = z \sqrt{\frac{\hat{p}(1 - \hat{p})}{N}}\text{.}\)
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βΈ This calculation should be used for Monte Carlo simulations involving binary outcomes (βhitβ vs βmissβ, βyesβ vs βnoβ).
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Helpers:
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z_from_CL
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Examples:
% Example 1: Margin of error for pHat = 0.6, N = 1000
ME = pHat_marginErr_w_CL(0.6, 1000)
% Returns ME = 0.0304
% Example 2: 99% confidence level
ME = pHat_marginErr_w_CL(0.25, 500, 0.99)
% Returns ME = 0.0499
Subsection muHat_marginErr_w_CL
This helper function computes the margin of error for a sample mean estimate \(\hat{\mu}\) based on the sampleβs standard deviation and a desired confidence level.
Inputs:
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s (1x1) double β estimated sample standard deviation.
N (1x1) integer β number of samples or trials.
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Outputs:
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Details:
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βΈ The margin of error formula is \(\text{ME} = z \frac{s}{\sqrt{N}}\text{.}\)
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βΈ This calculation should be used for continuous-valued Monte Carlo experiments where each trial produces a numeric measurement.
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Helpers:
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z_from_CL
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Examples:
% Example 1: Margin of error for s = 0.5, N = 100
ME = muHat_marginErr_w_CL(0.5, 100)
% Returns ME = 0.0980
% Example 2: 99% confidence level
ME = muHat_marginErr_w_CL(0.1, 1000, 0.99)
% Returns ME = 0.0081
Subsection mc_prob_H_heads_F_flips
This function uses Monte Carlo simulation to estimate the probability of obtaining exactly \(H\) heads when flipping a fair coin \(F\) times.
Inputs:
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H (1x1) integer β desired number of heads.
F (1x1) integer β number of coin flips per trial.
seed (1x1) integer β optional rng seed for reproducibility.
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Outputs:
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Details:
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βΈ Use randi([0 1],*,*) to create a random row vector of ones and zeros, where 1 = heads and 0 = tails.
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Helpers:
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pHat_marginErr_w_CL
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Examples:
% Example 1: Estimate P(5 heads in 5 flips)
simEstimates = mc_prob_H_heads_F_flips(5, 5, 5000)
% simEstimates =
%
% struct with fields:
%
% N: 5000
% prob: 0.0324
% ME_CL95: 0.0049
% Example 2: P(3 heads in 2 flips) = 0 (Impossible)
simEstimates = mc_prob_H_heads_F_flips(3, 2, 100)
% simEstimates =
%
% struct with fields:
%
% N: 100
% prob: 0
% ME_CL95: 0
% Example 3: P(10 heads in 13 flips) with seed=33
simEstimates = mc_prob_H_heads_F_flips(10, 13, 1000, 33)
% simEstimates =
%
% struct with fields:
%
% N: 1000
% prob: 0.0410
% ME_CL95: 0.0123
Subsection mc_approx_pi
This function uses Monte Carlo simulation to estimate the value of \(\pi\text{.}\)
Inputs:
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seed (1x1) integer β optional rng seed for reproducibility.
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Outputs:
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simEstimates (1x1) struct containing the fields:
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Details:
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βΈ To code this, slightly alter the in-class demonstration where we approximated the area of a circle of radius r. Think about the connection of \(\pi\) to this demo.
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Helpers:
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pHat_marginErr_w_CL
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Examples: Note: since no seed was used in Example 2 your values will be different.
% Example 1: N=10^5 with seed
results = mc_approx_pi(5000, 10)
% results =
%
% struct with fields:
%
% approxPi: 3.1680
% N: 5000
% pHat: 0.7920
% ME_CL95: 0.0113
% Example 2: No inputs, using the default N and no seed
approx_pi = mc_approx_pi()
% results =
%
% struct with fields:
%
% approxPi: 3.1422
% N: 100000
% pHat: 0.7855
% ME_CL95: 0.0025
Subsection mc_prob_die_roll_sum
This function uses Monte Carlo simulation to estimate the probability that the sum of \(D\) fair six-sided dice equals a specified target value.
Inputs:
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D (1x1) integer β number of dice rolled per trial.
seed (1x1) integer β optional rng seed for reproducibility.
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Outputs:
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Details:
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βΈ Use randi to generate a random 1x6 roll.
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Helpers:
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pHat_marginErr_w_CL
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Examples:
% Example 1: Estimate P(sum = 7) for two dice
simEstimates = mc_prob_die_roll_sum(2, 7, 10000)
% simEstimates =
%
% struct with fields:
%
% N: 10000
% prob: 0.1658
% ME_CL95: 0.0073
% Example 2: Reproducible estimate with fixed seed
simEstimates = mc_prob_die_roll_sum(3, 10, 2e6, 42)
% simEstimates =
%
% struct with fields:
%
% N: 2000000
% prob: 0.1254
% ME_CL95: 4.5891e-04
Subsection mc_sum_until_S
This function using Monte Carlo simulation to estimate the expected number of uniformly distributed random samples (between 0 & 1) required for their cumulative sum to exceed a target value.
Inputs:
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target_sum (1x1) double β threshold for the running sum of draws.
seed (1x1) integer β optional rng seed for reproducibility.
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Outputs:
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Details:
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βΈ Use rand() to generate the random numbers between 0 and 1.
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Helpers:
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muHat_marginErr_w_CL
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Examples:
% Example 1: Expected draws to reach sum β₯ 1
simEstimates = mc_sum_until_S(1, 100000)
% simEstimates =
%
% struct with fields:
%
% N: 100000
% avgSamples: 2.7126
% ME_CL95: 0.0054
π€π What does this average π appear to be close to?
% Example 2: Reproducible run with fixed seed
simEstimates = mc_sum_until_S(3.0, 200000, 31415)
% simEstimates =
%
% struct with fields:
%
% N: 200000
% avgSamples: 6.6642
% ME_CL95: 0.0065
Subsection mc_prob_same_bday
This function uses Monte Carlo simulation to estimate the probability that, in a group of \(P\) people, at least two share the same birthday.
Inputs:
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P (1x1) integer β number of people in the group.
seed (1x1) integer β optional rng seed for reproducibility.
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Outputs:
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Details:
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βΈ Each trial simulates a group of P people where each person has random birthday (integer from 1 to 365, assuming no leap years).
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βΈ Use randi to generate the random birthdays.
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Helpers:
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pHat_marginErr_w_CL
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Examples:
% Example 1: Estimate probability for 23 people (the classic case)
simEstimates = mc_prob_same_bday(23, 100000)
% simEstimates =
%
% struct with fields:
%
% N: 100000
% prob: 0.5081
% ME_CL95: 0.0031
% Example 2: Reproducible simulation with fixed seed
simEstimates = mc_prob_same_bday(30, 500000, 123)
% simEstimates =
%
% struct with fields:
%
% N: 500000
% prob: 0.7075
% ME_CL95: 0.0013
Subsection mc_prob_dice_poker_rolls
For this problem, you will need the following dice poker helper functions you wrote in Homework 3 and 4:
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get_face_counts,sort_by_rank -
is_5Kind,is_4Kind,is_straight, ... ,is_1pair
Make sure the programs, below, can see them in you working folder.
This function uses Monte Carlo simulation to estimate the probability of the eight possible 5-dice poker hand ranks.The estimates are summarized as confidence interval strings for each hand rank.
Inputs:
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seed (1x1) integer β random number generator seed.
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Outputs:
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simEstimates (1x1) struct containing the following fields:
where each confidene interval is formatted as β\(pΜ%\) Β± \(ME%\)β.
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Details:
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βΈ Hard code the confidence levels to 95% and use your pHat_marginErr_w_CL to find the margin of error for each estimate.
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βΈ Results are formatted as percentages with one decimal place for pHat and two decimals for ME. Use round to set the decimals.
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Helpers:
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Examples:
% Example 1: Run a default simulation
simEstimates = mc_prob_dice_poker_rolls()
% Returns:
%
% struct with fields:
%
% fiveKind: "0.1% Β± 0.02%"
% fourKind: "1.9% Β± 0.08%"
% straight: "3.8% Β± 0.12%"
% fullhouse: "3% Β± 0.11%"
% threeKind: "15.7% Β± 0.23%"
% twoPair: "23.1% Β± 0.26%"
% onePair: "46.1% Β± 0.31%"
% singles: "6.1% Β± 0.15%"
% Example 2: Reproducible results with fixed seed
simEstimates = mc_prob_dice_poker_rolls(500, 123)
% Returns:
%
% struct with fields:
%
% fiveKind: "0.4% Β± 0.55%"
% fourKind: "1.6% Β± 1.1%"
% straight: "4% Β± 1.72%"
% fullhouse: "2.6% Β± 1.39%"
% threeKind: "16.8% Β± 3.28%"
% twoPair: "22.8% Β± 3.68%"
% onePair: "43.8% Β± 4.35%"
% singles: "8% Β± 2.38%"
